Circle theorems explained: rules, examples and reasons
Learn the main circle theorems and use them to find angles with a reason for every step. GCSE Higher and high school Geometry, with worked examples.
GCSEA-LevelHigh SchoolCurriculum fit: GCSE Maths Geometry and measures (Higher tier). US: Geometry (Circles).
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What is it?
Circle theorems are rules about angles and lines in circles. Exam questions usually ask you to find an angle and to state the theorem that justifies each step.
You need to know the key vocabulary: centre, radius, chord, tangent, arc, segment, and cyclic quadrilateral (a four-sided shape with all vertices on the circle).
Key formulas and ideas
- The angle at the centre is twice the angle at the circumference (on the same arc).
- Angles in the same segment are equal.
- The angle in a semicircle is 90°.
- Opposite angles of a cyclic quadrilateral add to 180°.
- A tangent is perpendicular to the radius at the point of contact.
- Two tangents drawn from the same point are equal in length.
Worked examples
Example 1: Centre and circumference
Question: The angle at the centre of a circle is 140°. Find the angle at the circumference on the same arc.
- The angle at the centre is twice the angle at the circumference.
- 140° ÷ 2 = 70°.
Answer: 70° (angle at centre is twice angle at circumference)
Example 2: Cyclic quadrilateral
Question: One angle of a cyclic quadrilateral is 85°. Find the opposite angle.
- Opposite angles of a cyclic quadrilateral add to 180°.
- 180° − 85° = 95°.
Answer: 95°
Example 3: Semicircle
Question: AB is a diameter of a circle and C is on the circle. Angle CAB = 35°. Find angle ABC.
- The angle in a semicircle is 90°, so angle ACB = 90°.
- Angles in a triangle add to 180°: 180° − 90° − 35° = 55°.
Answer: 55°
Common mistakes
- Giving no reason, or a reason that is too vague.
- Using the centre-angle rule on the wrong arc.
- Confusing the tangent-radius angle, which is 90°, with other angles.
- Forgetting that the rule for cyclic quadrilaterals applies to opposite angles only.
Practice questions
Frequently asked questions
What is the alternate segment theorem?
The angle between a tangent and a chord equals the angle in the alternate segment. It is a Higher tier theorem.
How do I remember the theorems?
Draw a small sketch for each one and learn the key phrase, such as 'angle at the centre is twice the angle at the circumference'.
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GCSEHigh SchoolSources and review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.